For the Stuart–Landau model, we can obtain a phase function \(\theta _i\)  and a PCF \(\Gamma _i\)  for \(i=1,\ldots ,N\)  analytically from the original model [1]}, [2]}.
In Figs. REF  B and C, we show the phase functions globally, which are
obtained from the original model and the estimated Koopman operator, respectively.
Moreover, Figs. REF  D–G show the difference of the PCF \(\Gamma _d\)  obtained by the model-based approach and the estimated Koopman operator.
Regarding the estimation of the function \(\Gamma _d\)  with the Koopman operator, we set the value \(M\)  in Eq. (REF ) as \(1\sim 4\) .
Since the true PCF of the Stuart–Landau model obtained analytically is represented only with a sine function, setting \(M\)  as 1 is sufficient for obtaining an accurate estimation.
Therefore, the estimated function does not change even if \(M\)  is set as a larger value than 1 in this example.
